Computation of some parameters of Lie geometries

Computation of some parameters of Lie geometries

Citation for published version (APA):

Brouwer, A. E., & Cohen, A. M. (1983). Computation of some parameters of Lie geometries. (Mathematisch Centrum. ZW, afdeling zuivere wiskunde; Vol. 198). Stichting Mathematisch Centrum.

Document status and date: Published: 01/01/1983

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mathematisch

centrum

AFDELING ZUIVERE WISKUNDE

(DEPARTMENT OF PURE MATHEMATICS)

A.E. BROUWER

&

A.M. COHEN

ZW 198/83

COMPUTATION OF SOME PARAMETERS OF LIE GEOMETRIES

~

Me

SEPTEMBER

by

A.E. Brouwer

&

A.M. Cohen

ABSTRACT

In this note we show how one may efficiently compute the parameters of a finite Lie geomet,ry and we give the results of such computations in the most interesting cases. We also prove a little lemma that is useful for

showing that thick finite buildings do not have quotients which are (locally) Tits geometries of spherical type.

A

geometry

over a set Il (the set of

types)

is a triple (r,*,t) where r is a set (the set of

objects

of the geometry), * is a symmetric relation qn r (the

incidence

relation) and t is a mapping (the

type

mapping) from r into Il, such that for x,y E r we have (t(x)=t(y) A x*y) ~ x = y.

[An example is provided by the collection r of all (nonempty proper) subspaces of a finite dimensional projective space, with t:

r

+ 6 = :N the rank function, and * symmetrized inclusion (i.e., x*y iff x

£

y or y £ x).J

Often we shall refer to the geometry as r rather than as (r,*,t).

A

fZag

is a collection of pairwise incident objects. The

residue

Res(F) of a flag F is the set of all objects incident to each element of F. Together with the appropriate restrictions of * and t, this set is 'again a geometry.

The

rank

of a geometry is the cardinality of the set of types 6. The

corank

of a flag F is the cardinality of 6\t(F). A geometry is

connected

if and only if the (looped) graph (r,*) is connected. A geometry is

residuaZZy connected

when for each flag F of corank 1, Res(F) is nonempty, and for each flag of corank at least 2, Res(F) is nonempty and connected •

. :.

A

(Buekenhout-Ti"ts) diagrcun

is a picture (graph) with a node for each element of 6 and with labelled edges. It describes in a compact way a set of axioms for a geometry

r

with set of types Il as follows: whenever an edge

(d

Id2) is labelled with

V,

where

V

is a class of rank 2 geometries, then each residue of type {d

I,d2} of r must be a member of

V.

(Notice that a residue of type {d

I,d2} is the residue of a flag of type 6\{d1,d2}.) In the following we need only two classes of rank 2 geometries. The first is the class of all projective planes, indicated Ln the diagram by a plain edge. The second is the class of all generalized digons, that is, geometries with objects of two types such each object of one type is incident with

every object of the other type. Generalized digons are indicated in the diagram by an invisible (i.e., absent) edge.

o-o~

is an axiom system characterizing the geometry of points, lines and planes of projective 3-space. Note that the residue of aline (i.e., the points on the line and the planes containing the line) is a generalized digon. Usually, one chooses one element of ~ and calls the objects of this type

points.

The residue of this type are called

Zines.

Thus lines are geometries of rank I, but all that matters is that they constitute subsets of the point set. In the diagram the node corresponding to the points is encircled.

As an example, the principle of duality in projective 3-space asserts the isom~rphism of the geometries

0-0-0

and

0-0-0·

Grassmannians are geometries like

0-0-0·

(Warning: points are objects of the geometry but lines are sets of points, and given a line, there need not be an object in the geometry incident with the same set of points.)

Let us write down some diagrams (with nodes labelled by the elements of ~)

for later reference.

A: n D : n

0-0-0-

----0

1 2 3 n

o

()---()-()- --6-D

_{1 2 3 n-2 n-l}

0

6

0-0-0-0-0

_{1 2 3 4 5}

E7:

0-0-0

-0-0

1 2 3 5 6

8

E8:

0-0·-· 0

-0-0

1 2 3 4 5 6 7

(Warning: in different papers different labellings for these diagrams are used. )

If one wants to indicate the type corresponding to the points, it is added as a subscript. E.g. D4 1 denotes a geometry belonging to the diagram

_,

@~~.

1 2 3

One may prove than if

r

is a finite residually connected geometry of rank at least 3 belonging to one of these diagrams having at least three points on each line then the number of points on each line is q + 1 for some prime power q, and given a prime power q there is a unique geometry with given diagram and q + 1 points on each line. We write X (q) for this unique

n

geometry, where X is the name of the diagram.

n

[For example, A (q) is the geometry of the proper nonempty subspaces

n

of the projective space PG(n,q). Similarly, D (q) is the geometry of the

n

nonempty totally isotropic subspaces in PG(2n-l,q) supplied with a

non-degenerate quadratic form of maximal Witt index. Finally. D .(q) is an _{-. n,l} -example of a polar space.]

2. DISTANCE DISTRIBUTION DIAGRAMS FOR ASSOCIATION SCHEMES

An association scheme is a pair (X,{RO, ••• ,R

s}) where X is a set and the R. (O~i~s) are relations on X such that {RO, ••• ,R } is a partition of

~ . s

(i)

(ii)

RO

=

I, ViHI the identity T R. = R'I. 1. 1. relation.

(iii) Given x,y E X with (x,y) E R., then the number 1.

and (y,z) E ~} does not depend on x and y but

Pjk

=

#{z

I

(x,z) E R

j only on i.

The obvious example of an association scheme is the situation where a group G acts transitively on a set X. In this case one takes for {RO, .•• ,Rs} the partition of X x X into G-orbits, and requirements (i)-(iii) are easily verified.

Assume that we have an association scheme with a fixed symmetric nonidentity relation R} (i.e.,

R~

=

R

t). Clearly (X,R}) is a graph.

Now one may draw a diagram displaying the parameters of this graph by drawing a circle for each relation R., writing the number k. = #{z

I

(~,z) E R

i}

=

o 1. 1.

P .. where x _1.1. E X is arbitrary inside the circle, and joining the circles for

i 1.

R. and R. by a line carrying the number p .. at the (R.)-end whenever PJ'}.

f

O.

1. J' • k . J1 1. .

(Note that kiopjI

=

kj"Pil so that Pjl is nonzero iff

pi}

is

n~nzero.)

When i = j, one usually omits the line and J'ust writes the number p next to

it

the circle for R.o 1.

For example, the Petersen graph becomes a symmetric association scheme i.e., one for which

R~

=

R. for all i when we define (x,y) E R.

~

d(x,y)

=

1.

1. 1. 1.

for i 0,1,2. We find the diagram

(0~.

3 I 2 t 2

More generally, a graph G is called

distance regular

when (x,y) E R. ~ 1. d(x,y)

=

i (O~i~diamG) defines an association scheme.

When (X,R₁) is a distance regular graph, or, more generally, when the matrices

') ~

I,A,A~, •• ",A~ are linearly independent (where ~ is the O-} matrix ofR

t, i.e.,

t~e adjacency matrix of the graph), then the Pj} s~ffice to determine all

Pjk· On the other hand, when the association scheme is not symmetric but R) is, then clearly not all R

j can be expressed in ter~s of RIO

In this note our a1.m is to compute the parameters Pjk for the Lie geometries X _m,n (q) where X is a (spherical) diagram with designated 'point'- type,n, _m and the association scheme structure is given by the group of (type preserving) automorphisms of X (q) - essentially a Chevalley group. In the next section

we shall give formulas valid for all Chevalley groups and in the appendix we list the results in some of the more interesting cases. Let us do some easy examples explicitly. (References to words in the Weyl group will be explained in the next section.)

i

Usually we give only the Pjl; the general case follows in a similar way. EXAMPLE I.

An,1

@-Q-O"'O'

1 2 3 n

The col linearity graph of points in a projective space is a clique: any two points are adjacent (collinear). Thus our diagram becomes

~

Vk

I\..V

k-I EXAMPLE 2. A n,2

0--0-0"'0

I 2 3 n n q -I k = - - 1 .q q- v - I .

Now we have the graph of the lines in a projective space, two lines reing adjacent whenever they are in a common plane (and have a point in common).

[N.B.: the Lines of this geometry are pencils of q + lines in a common plane and on a common point.]

projectiv.e

Our diagram becomes

'1 n-2 .' ~n -1 Aq-·"q-I-Weyl words: "" "2" "2312" v

₌

k

₌

A

₌

k2

=

n+1 n (q -I)(q -I) 2 (q -I)(q-I) n-I q -I q(q+l) _q-1 2 2 n-2_ 1 q-I+q +q .q q-I n-I -I n-2 -I q q 2 q -I q-I 4 q

.

For q = 1 (the 'thin' case) this is the diagram for the triangular graph:

o

Zn-Z

n-l i

[Clearly Ai := P1i = k - ~'4' Pl" i Often, when A.

JT~ J ~ does not have a particularly

nice form, we omit this redundant information.]

Notice how easily the expressions for v,k,kZ,A can be read off from the Buekenhout-Tits diagram: for example, A = A(X,y) first counts the q-l points on the line xy, then the remaining qZ points of the unique plane of type {1,2} containing this line and finally the remaining q2 points of the planes of type {Z,3} containing this line.

EXAMPLE 3.

An,j

0-0- ...

--Q)- ..

·0

2 j n

This is the graph of thej-flats (subspaces of dimension j) in projective n-space, two j-flats being adjacent whenever they are in a common (j+l)-flat

(and have a (j-l)-flat in common). The graph is distance regular with

\'

diameter j. Parameters are:

iZ [j] [n-j+l]

= q • . • • •

~ q 1, q

b '= i _{. , Pl' 1 q} = Zi+l [j-i] [n-j-i+l] _,

~ ~+ q q

The parameters for the thin case have q = I and binomial instead of Gaussian coefficients; we find the Johnson scheme (n;l).

J

The Weyl words (minimal double coset representatives in the Weyl group) have the following shape: for double coset i in A . the representative ~s

n,J w. ="j,j+l, j+Z, ... , j+i-l, j-l, j, j+l,

~

...

,

j+i-Z,

j-i+l, j-i+2, ••• , j".

Note that wi has length i 2, the power of q occurring in k i . EXAMPLE D n,1

4.

~

(Q)-O- ...

-0"-0·

2 n-2 n-l (n~3; D

2,1 is the direct productA1,1 x A1,1' i.e., a (q+l) x (q+l) grid.)

Diagram: Thin case: v

=

#D n,1

=

_{(qn_ 1)(qn-l+ 1)} q-l k = q.#Dn- 1, I

8-k---.-.

~-q"-2-n--_3_-#D-n---l....:,'-;le

q-l+q .#Dn- 2 ,1 (q-1).#D_n- ₁,1 v

=

2n, k

=

2n-2 This is K

2n minus a complete matching.

The Weyl words are: ""for double coset 0, "1" for double coset 1, "123 •.. n-3 n-2 n n-l n-2 ••• 1" for double coset 2.

EXAMPLE 5.

--3-0

(n ~

5).

Dn,2

0-0-1 2 n-2 n-l v #D .#D n,1 n-l, 1 #A 1 1

,

n n-l n - l ' n-2 (q -1)(q +l)(q -1)(q +1) 2 (q -I)(q-l) n-2 n-3 k

=

q.#A 1 ,1 .#D n-2, 1

=

q(q+l) . (q -l)(q q-l +1)

Diagram (for n > 4): ( q+ 1 )2 2n-6 .q 1

\.:...Ik

q.#Dn-3

,

1 p2_n_-_4 ________

k_/~

"-...-/

(Double coset 1 contains adjacent points, i.e., lines of the polar space in a common plane. Shortest path in the geometry: 2-3-2 (unique).)

Double coset 2 contains the points at 'polar' distance two, belonging to the Weyl word "2312", l..e., in a polar space A3 2' (I.e., lines or the polar

_,

space in a common t.i. subspace.) Thus

n-2

q -1 n-4 4

• (q + 1) • q • q-I

Shortest path in the geometry: 2-4-2 (unique). Double coset 3

points incident with a common I-object, so that 'the Weyl word is the one for double coset 2 in Dn-I,I (relabelled):

"23 ••• n-3 n-2 n n-I n-2 ••• 2".

(These are intersecting lines not l.n a common t.i. plane.) Thus

Shortest path in the geometry: 2-1-2 (unique).

Double coset 4 contains points with shortest path 2-1-3-Z (unique); the Weyl word is

"z

3 ••• n-3 n-Z n n-I n-2 ••. 3 1 Z",

the reduced form of the product of the word we found for double coset 3 and the word "ZI2'.' des crib ing adj acency in A

Z

,

Z. Thus # 2 # Z # k4 = _{Dn-Z,l' q • (Dn- 1,1-(q+I)-q • Dn- 3},1) n-Z q -1 n-3 Zn-3

=

.

(q +1). (q+l) • q q-l 4n-7

Double coset 5 contains the remaining q points (the lines of the polar space in general position). Shortest path in the geometry: 2-1-2-1-2 (not unique). The Weyl word is

" Z 3 ••• n-) 1 Z ••• n-Z n n-Z ••• Z 1 n-l ••• 3 Z " of length 4n-7.

The thin case is:

v = Zn(n-l),

k

4

(n-Z) Z(n-3) 4 Z(n-3)

4

(n-Z) 2(n-Z)

EXAMPLE 6. 2 3 As before we find 6 4 6 q -I q -1 2 __ q -1 2 2 v = - 2 - • • (q + 1 ) • (q + 1 ) and k q -I q-I q-I

This time the thin diagram is

v

=

24, k 8

3 q(q+l) •

and we see that the number of classes is one higher than before. This is "

caused by the fact that we can distinguish here between shortest paths 2-4-2 and 2-3-2, while in the general case (n ~ 5) both 2-n-2 and 2-(n-I)-2 are equivalent to 2-3-2. Thus, our previous double coset 2 splits here into two halves.

Double coset Weyl word Cardinality Shortest path (unique)

0

'"'

1 2 "2" q(q+l) 3 2-{1,3,4}-2 2 "231~" q (q+l) 4 2-4-2 3 "2412" q (q+l) 4 2-3-2 4 "2432"·· q (q+ 4 1) 2-1-2 5 "24312" . q (q+l) 5 3 2-1-{3,4}-2 6 "231242132" q 9

Diagram EXAMPLE 7. D n,n

0-0-1 Z Z (q+ 1)

This graph LS distance regular of diameter [~nJ.

We have n~l n-2 k

q[~Jq'

v = (q +l)(q +1) .•• (q+I), (Zi) = 4i+l[n-Zi_J Z n k. q . ~.J, b. _{q Z}

_q'

c. = L Lq L L [Zzi Jq • m(Zm-l)

Note that when n

=

Zm, then k

=

q • Also, that in case n = 4 these

m

parameters reduce to those we found for D4 1.

,

'

Two points have distance ~ i (for O~i<n) iff there is a path n-(n-Zi)-n in the geometry. When n is even then two points at distance ~n ("in general position") are not incident to a conrrnon object. (Note that k

=

#A -I Z. q

_ _ i(Zi-l) n ,

and, more generally, that k. - #A

n_1 Z· . k.(DZ· Z·) - q • #A 1 Z··

L ,L 1 L, L n- , L

The values for b. and c. follow similarly. The value for v follows by

1 L

induction, and when n

=

Zm then k is found from k

=

v

-m m l:. L<m k .• ) L

The Weyl word corresponding to distance i is the same one (after relabelling) as in D_Z· Z., namely:

"n n-2 n ... I n-3 n-2 n n-4 n-3 n-2 n-I

_"

of length

I + 2 + 3 + 4 + ••• + 2i - I

=

i(2i-l).

n-I n .

In the thin case we have v = 2 , 'k = (2)' and the graph is that of the binary vectors of even weight and length n where the distance is the Johnson distance, i.e., half the Hamming distance.

EXAMPLE 8 (See Tits [8J). 6

QJ--O

-0-0

I 2 4 5

This graph is strongly regular (i.e., distance regular with diameter 2). We have 12 9 8 3 q -I _~ and #Dr:. q -I v = 4 k q

.

r:. q

.

.. (q+l). q -1 q-l - ' , - ' q-l The thin case gives diagram

v

=

27

0.:-B~

161 ₁₀5 88

the Schlafli graph - this is the complement of the collinearity graph of the generalized quadrangle GQ(2,4). In general we find diagram

O

. , - - - k··

1~_7.#A41

, k

# 2

k A D4 ,1

8 2

where k2

= q .#D

₅,I and

A

= q-l + q .#A

₄,2"

Double coset corresponds to shortest path 1-2-1 and has Weyl word "I". Double coset 2 corresponds to shortest path 1-5-1 and has Weyl word ,. 12364321 ", as in D

EXAMPLE 9.

Q--D

1 2

This graph has

and 9 1 v=~ q-l

0-0

3 4 5 (q6+ 1)(q4+ 1)(q3+ 1) 5 . 2 .. _ 3 _ . Q -1 q (q - + 1 H q - + I )-' -1 •

q-The thin case gives diagram

v = 72, 8 In general we find 1 t-k _ _ _ _ - - f • . • #' Jl. 6.11

la,

A1~~_1 ... 2 (21'12 w~tn 1.<.2

=

8.

5,1. 118.4,1. q anu 1:\.3 = q I:\. anu 1\ - ' i • • ' i • ,q +q+./ •

Double coset corresponds to shortest path. 6-3-6 and has Weyl word "6". Double coset 2 corresponds to shortest path 6~{1,5}-6 and has Weyl word "634236" (of D4 1).

_,

Double coset 3 corresponds to shortest path 6-1-4-6 (or, equivalently~

6-5-2-6) and has Weyl word "6345 234 1236".

Double coset 4 has Weyl word "6345 234 1236345 234 1236". For examples of type F₄,1' see Cohen [6J.

Up to now all our computations were easy and straightforward, mainly because of the limited permutation ranks (number of classes of these association schemes) and the fact that A l' D 1 and E6 1 have diameter at most two.

n, n, ,

Continuing in this vein we quickly encouter difficulties. E

7,1 is still distance regular with diameter 3 and E

7,6 and ES,l have diagrams like E₆,6 (and these three cases are easily doable by hand) but for instance E

7,4 has 149 classes (double cosets) and all geometric intuition is lost; in the next section we describe how the parameters for these Lie geometries can be mechanically derived by means of some computations in the Weyl group. In a way, this means that it suffices to consider the case q

=

1. Now

everything is finite and a computer can do the work.

In the appendix we give computer output describing E₇

,1'

_{E7,6' E7,7' ES,l'}

ES 7 ,and ES S' in other words, the geometries belonging to the 'end nodes'

_,

_,

of the diagrams E7 and ES' For E7 we also computed the parameters on the remaining nodes, but listing these would take too much room. We therefore content ourselves with the presentation of permutation ranks for the Ghevalley groups of type F 4' En (6::;n::;;S); to each node r in the diagrams below is attached the permutation rank of the Cheval ley group of the relevant type on the maximal parabolic corresponding to r.

5 17 17 5

0

5

0-0-6-0-0

3 10 37 10 3

QIO

0-0-0-6-0-0

4 13 50 149 27 5

0-0

_{5 26}

3. REDUCTION TO THE WEYL GROUP

In this section,G is a Chevalley group X (q) of type X over a finite

n n

fieldW • We shall heavily rely on Carter

[4J,

to which the reader is

q

referred for details. Though with a little more care, all statements can be adapted so that they are valid for twisted Chevalley groups, too, for the sake of simplicity, we shall only consider the case of an untwisted Cheval ley group G. To G we can associate a split saturated Tits system

(B,N,W,R), cf. Bourbaki [IJ, consisting of subgroups B, N of G such that G is generated by them, and of a Coxeter system (W,R) with the following properties

(i) H

=

B n N is a normal subgroup of Nand W =

NIH.

(ii) For any w E Wand r E R

(ii)' BwBrB £ BwB u BwrB (ii)" rB.£ B

(iii) (split) There is

w

a normal subgroup U of B with B = UH and U n H

=

{I}.

(iv) (saturated) n B = H.

W WEW 1

Here and below, A stands for wAw- if A is a subset of G invariant under conjugation by H. Notice that wB and Bw are well defined. We shall briefly recall how the Tits system may be obtained. Start with a Coxetersystem

(W,R) where W is a Weyl group of type X • Let ~ be a root system for W. A

n

set of mutually obtuse roots corresponding to the subset R (of

fundamentaZ

reflections)

forms a set of

fundamental roots.

Now, any root ~ E ~ is an

integral linear combination of the fundamental roots such that either all coefficients are nonnegative or all coefficients are nonpositive. In the former case ~ is called

positive,

notation ~ > 0, in the latter case ~

is called negative, notation ex. ~ ... v. n

Now choose a Cartan subgroup H in G, and denote by Xfor~ E ~ the

~

root subgroup with respect to ~ (viewed as a linear character of H). Thus H normalizes each X • Next, let N be the normalizer of H in G. Then W =

NIH

~ .

permutes the X (~E~) according to Wx = X (WEW).

~ ~ ~

Now U

=

TIO X is a subgroup of G normalized by H, so that B ~ UH is a a.> ~

subgroup of G with B n N = H. This explains how B,N, W,R, U occur in G. We need some more subgroups of G. Given w E W, set

U

w := ~>O II w-l~<O

X .

This is a subgroup of U. In fact, if lew) for w E W denotes the length of w with respect to R, there is a unique longest element Wo in

W;

this element Wo is an involution satisfying U n woU = 0{1}, and U: = U n wwoU•

Notice that our definition of U- differs from Carter's in that our U

w oW

coincides with his U--l' It is of crucial importance to the computations

w

below that

.e.

(w)

=

q , for every W E

W.

Fix r E Rand write J = R\{r}, W

J

-

<J>, the subgroup of loJ generated by and P = BWJB. Then P is a socalled maximal parabolic subgroup of G

oL -

.

(associated with r) • We are interested

in

the graph r = r(G,p) defined as follows:

its vertices are the cosets xP in G (for x E G), two vertices xP, yP being

-1

adjacent when y x E PrP.

In this graph, xP and yP have distance d(xP,yP) ~ e if and only if

-1

Y x E P <r> P <r> ••• <r> P (a product of 2e + terms). Let us first compute the number v of vertices of this graph.

LEMMA 1.

Eaah aoset

xP

has a unique representation

xP = uwP

where

u E U

w

and

w

is a right J-reduaed element of W, i.e.,

W E L

J := {w E W Il(ww') ;:: lew)

for aU

w' E WJ}.

PROOF. xB has a (unique) representation xB = uwB with w E W, U E U , see

w

Carter [4J, Thm. 8.4.3. Thus xp = uwP and obviously we may take w E L

J (cL Bourbaki Ci3J, Exercice 3, §l). Suppose uwP=u'w'P. Then w' E B"lBWJB so that

w' =ww" with W"EW

J, but since w,w' (SoLJ it fol1.ows that w' =w. Next, slnce

-1 -I - Wo w

Pnw Bw.s.B and w Uww.s. U and l):n °U= I (cL [S}, Proposition p. 63,

Carter [4J, Lemma 7.1.2), it follows that u = u'.

D

PROPOSITION I.

The graph

r(G,p)

has

v

vertiaes, where

v =

L

q lew) •

wEL J

PROOF. A str~ightforward consequence of the formula Iu-I

w

and Lennna 1.

D

REMARK I. Of course, we also have the multiplicative formula v n II

i=1

d' e'

(q 1-1)/(q 1-1),

where dI, .•• ,d

n are the degrees of the Weyl group W, e2, ..• ,en are the degrees of the Weyl group W

J, and el

=

I, (cf. Carter [4J).

Next, we want to put the structure of an association scheme on this graph. The group G acts by left multiplication on the cosets xP, and clearly this action is transitive. Tnus we find an association scheme. TIle

collections of co sets in a fixed relation with a given coset, say P, are the double cosets PxP" The pair (xP,yP) has relation G(xP,yP), labelled

-1 -I

with Px yP. We see that a relation PxP is symmetric iff PxP

=

PxP, and this holds in particular for x

=

r.

LEMMA 2.

Each double coset

PxP

has a unique representation

PxP = PwP

where

w

is a both left and right J-reduced element of

W~ i.e.~

W E D

J := {w E W

I

w

is the unique shortest word of

WJWWJ}. PROOF. See Bourbaki [I

J

Chap. IV § 1 Exercic e 3.

0

PROPOSITION 2.

The association scheme

r(G,p)

has valencies

k.

(belonging

1.

to relation

PiP)

for

i E D

J,

where

L

k. 1. wELJnWJi PROOF. Obvious.

0

lew)

q •

REMARK 2. If i E D_J, then iW_Ji-1 n W_J

=

WiJi-lnJ by Solomon [7J, so substitution of q

=

1 in the above formula for k. leeds to the equation

1.

1. Finally, we come to the pa,rameters P

jk, It is 'more convenient to label the relations (such as i,j,k) by elements from D

J than by 0,1, ••• ,8 as in Section 2. Therefore, we shall use these new labels; 1 now stands for' the "old 0", an~ r for adjacency, i.e., the "old 1". We shall confine ourselves

. . . 1. to g1 V1.ng p. •

THEOREM 1.

Let

i,j E D

J•

Then the number of points

(i.e.~

cosets) in

iPrP n PjP

is

I

i·(w) +

I

q£.(wr) (q_.:

wELnA wELnA

£. (iw) >£. (iwr) £. (iw) <£. (iwr)

PROOF. Clearly,

Consequently,

iPrP

=

iBWJBrP

=

iBwJrP

=

u. iBwP. we:L

wELnAr

..e.

(iw) <£. (iwr)

Now we want to write each set iBv]P as a union of cosets uwP as in Lemma 1.

g -1 # .

For g E G and K a subgroup G defineK := gKg and K = K\{I}. It is

i -

-well known that for any u E W we have

if

£.(iu) = £.(i) + £.(u)

then

CU) cU . •

u ~u

(See Cohen [5J Lemma 2.11.) Notice that w = vr for some v e: W_J with £.(iv)

=

£.(i) + £.(v) and £'(vr)

=

£.(v) + 1.

Distinguish two cases: If £.(iw) > £.(iv) , then

iBwB

=

iu wB

=

~(U-)iwB

w w

i -

-and we have (U) < U. as desired.

w ~w

If £.(iw) < £.(iv) then

iBwB = iBvBrB ~ (U )ivBrB

-v

i - - i - iw -

-and we have (U) c U., (U). (U)

s.

U. as desired.

v . UJ v r ~v

(For the inclusion ~(U~) c U: note that v cannot change the sign of the

v

~w

root corresponding to r since v E Wj')

Now in order to count how many of the cosets uwP fall into a given double coset PjP we only need observe that uwP c PjP iff w E WjjW

J , and that distinct w E.L lead to distinct cosets iwp.

0

COROLLARY 1.

Given two vertices

x

1P, x2P

of

r

at mutual distance

d~

the

number of vertices at distance

d - 1

to

xl P

and adjacent to_ x

2P

is congruent

to

1

(mod

q)~

and the number of vertices at distance

d

to

xIP

and adjacent

to

x

2P

is congruent to -} (mod

q). Also~

the valency

k

is congruent to

0

{mod

q}.

PROOF. From "w E WJr -<=> few) ~ I" and the expression given for k

=

kr we see that k

=

0 (mod q). Next, from the previous theorem we obtain that

o(ir E + (q-l) (mod q)

where oCT) for a predicate T denotes J if T is true and 0 otherwise. Thus,

11 i 0 ( d ) i h' h . 1 ( d )

a p. are congruent mo q except p. w ~c ~s congruent - mo q

Jr ~r

and p! which is congruent 1 (~od q) - where i is defined by ir E WJiW J•

~r

Clearly d(P,iP)

=

d(P,iP) - 1.

0

REMARK 3. This corollary ~s motivated by Lemma 5 in [2J which is a crucial

step in the proof that

if

r

is finite and q > 1, then the building

corresponding to the Tits system (B,N,W,R) does not have proper quotients satisfying the conditions in [10J, Theorem 1. The above corollary shows that the conditions are satisfied for the Chevalley groups of type A , D

n n

or E (6~~8). For another application, see [3J. m

i

REMARK 4. It is possible to compute the parameters Pjk for arbitrary k in a similar way. Again one starts by writing iPkP as a disjoint union of the form iBwP. Next by induction on few) this is rewritten as a disjoint

union of cosets uvP, where u E U

v and v E 1J• As an algorithm this works perfectly well, but it is not so easy to give a simple closed expression

i for Pjk'

4. COMPUTATIDON IN THE WEYL GROUP

We shall briefly discuss the way in which several items in the Weyl group have been computed.

(i) The length function f.

function; all other computations can be done by general group theoretic routines. But given the permutation representation of the fundamental reflections on the root system q, and a product representation

w = s 1 . s2' •• sm (not necessarily minimal), we find lew) from lew) = #{a E q,

la

> 0 and wa < O}

(see e.g. Bourbaki [1] Chap. VI, §1 Cor. 2). (ii) Canonical representatives of the cosets wW

J•

Let ~ be the coroot perp_ndicular to all fundamental roots except the one corresponding to r. Then ~ has stabilizer W

J in W, and the images of ~ under Ware in 1 - 1 correspondence with the cosets wW J.

(iii) Equality in W.

Similarly, let p be the sum of all positive roots. Then wp

=

w'p iff w = w' •

(iv) Double coset representatives.

Given a suitable lexicographic and recursive way of generating the cosets wW

J' the first of these to belong to a certain coset WJwWJ will have w E D

J• All cosets in the same double coset are generated by premultiplying previously found cosets with reflections in J. However, the set D

J of distinghuised double coset representatives can be found without listing all single cosets wW

J: given w E DJ, one can determine all elements from D

J n wL, where L

=

LJ n WJr, by simply sieving all right and left J-reduced words from wL (compare (i». In view of the fact that W is generated by J u {r}, iteration of this process will eventually yield all of D

J (one can start with w = 1). We have done so for the Weyl groups of type F

4, E6, E7, E8• The cardinalities of D

J, Le. the permutation ranks, have been given above. REFERENCES

[1] BOURBAKI, N.,

Groupes et Algebres de Lie,

Chap. 4, 5 et 6, Hermann, Paris, 1968.

[2J BROUWER, A~E.

&

A.M. COHEN,

Some Remarks on Tits

Geo~etries~ to appear in Indagationes Math.

[3J BROUWER, A.E.

&

A.M. COHEN,

Local Recognition of Tits Geometries of

classical

type~ preprint.

[4J CARTER, R.W.,

Simple groups of Lie

type~ Wiley, London, 1972. [5J COHEN, A.M.,

Semisimple Lie groups from a geometric

viewpoint~ pp.

41-77 in: "The Structure of real semisimple Lie Groups" ,Cede T.H. Koornwinder), MC Syllabus 49, Math. Centre, Amsterdam,

1982~

[6J COHEN, A.M.,

Points and Lines -in Metasymplectic

Spaces~. Annals of Discr. Math. 18 (1983) 193-196.

[7J SOLOMON, L.,

A Mackey Formula in the Group Ring of a Coxeter

Group~

J. Algebra 41 (1976) 255-268.

[8

J

TITS, J.,

Les "formes ree

l

'Les

/I

des groupes de type

E

6, Semina ire Bourbaki, no. 162, feb. 1958, Paris.

[9J TITS, J.,

Buildings of Spherical Type and Finite

BN-pairs~ Lecture Notes in Math. 386, Springer, Berlin, 1974.

[10J TITS, J.,

A local approach to

buildings~ pp. 519-547 in:

The Geometric

Vein~ (The Coxeter Festschrift), ed. Ch. Davis et al, Springer, Berlin, 1982.

27 cosets .3 double cosets Sizes:

c: ()

[ 1 ] 1: (1) [16J 2: (12364321) [ 10] 1 q +'0**2 + q**3 + 2*q**4 + 2*q**5 + 2*q**6 + 2*q**7 + 2*q**8 + q**9 + q**10 + q**11 q**8 + q**9 + q**10 + q**11 + 2*a**12 + q**13 + q**14 + 0**15 + 0**16 Neighbours of 1: [16J a point in 0: q + q**2 + q**3 + 2*q**4 + 2*q**5 + 2*q**6 + 2*q**7 + 2*q**8 + q**9 + q**10 + q**11 Neighbours of a point in 1: 0: [1J 1 1: [10J -1 + q + q**2 + q**3 + 2*q**4 +

"

.

Co • [5J Neighbours 1: [ 8 ]

".

Co • [8J 2*q**5 + 2*q**6 + q**7 + q**8 q**7 + q**8 + q~*9 + q**10 + q**11 of a point in 2: 1 + q + q**2 + ~*q**3 + q**4 + q**5 + q**6 -1 - 0**3 + q**~ + q**5 + q**6 + 2*q**7 + 2*q**8 + q**9 + q*~10 + q**11

***** e6,6 ***** 72 ccsets 5 doubLe ccsets Sizes: 0: () [lJ 1: (6) [20J ~

.

c:. • 1 q + q**2 + 2*q**3 + 3*q**4 + 3*q**5 + 3*q**6 + 3*q**7 + 2*0**8 + q**9 + q**10 (634236) [30J q**6 + 2*q**7 + 3*q**8 + 4*0**9 + 5*q**10 + 5*q**11 + 4*q**~2 + 3*q**13 + 2*q**14 + q**15 3: (63452341236) [ 2 0 J q * * 1 1 + q**12 + 2*0**13 + 3*q**14 + 3*q**15 + 3*q**16 + 3*q**~7 + 2*Q**18 + Q**19 + q**20

4:

(634523412363452341236) [1J q**21 Neighbours of a point in

0:

1: [20J 0 + q**2 + 2*q**3 + 3*q**4 + 3*q**5 + 3*q**6 + 3*q**7 + 2*q**8 + q**9 + q**10 !\Ieighbours of 0: [ 1 J 1 : 2 : ...

.

""0 [9J [9J [1J a point in 1: 1 -1 + q + q**2 + 2*q**3 + 3*q**4 + 2*q**5 + 0**6 q**5 + 2*0**6 + 3*0**7 + 2*0**8 + 0**9 q**10 Ne1ghcours of a point in 2: 1: [6J 1 + a + 2*q**2 + q**3 + q**4 2: [8J -1 - q**2 + q**~ + 2*0**4 + 3*q**5 + 2*q**6 + 2*0**7 3: [6J q**6 + q**7 + 2~q**8 + 0**9 + q**10 r~ e i 9 h b 0 u r s o t 1: [1 J 2: [9J 3: ['3 J 4: [1J fl.' e ; 9 h b 0 U r S 0 f 3: [20J 4: [ 0 J a point in 3: 1 q + 2*q**2 + 3*~**3 + 2*0**4 + 0**5 -1 - q**2 - a**~ + q**4 + 2*q**5 + 3*q**6 + 3*q**7 + 2*0**8 + q**9 q**10 a point in 4: ., 1 + q + 2*q**2 + 3*0**3 + 3*0**4 + 3*q**5 + 3*q**6 + 2*q**7 + q**8 + q**9 -1 - q**2 - a**¥ + Q**7 + q**8 + Q**10

***** e7,1 ***** 56 cosets 4 aouble cosets Sizes: 0: () [ 1 ] 1: (1) [27J 2: (1234?5't321) 1 q +., q * * 2 + q * *:3 + q * * 4 + 2 * 0

*

* 5 + 2*q**6 + 2*q**7 + 2*q**8 + 3*q**9 + 2*q**10 + 2*q**11 + 2*q**~2 + 2*0**13 + a**14 + 0**15 + q"*16 + a**17 [27] q**10 + q**11 + q**12 + q**13 + 2*q**14 + 2*q**15 + 2*q**~6 + 2*q**17 + 3*q**18 + 2*0**19 + 2*q**20 + 2*q**~1 + 2*q**22 + q**23 + q**24 + q**25 + q**26 3: (123475645347234512347654321) [lJ q**27 Neighbours of a point in 0: 1: [27] Q + q**2 + q**3 + Q**4 + 2*q**5 + 2*q**6 + 2*q**7 + 2*Q**8 + 3*q**9 + 2*q**10 + 2*q**11 + 2*q**~2 + 2*q**13 + Q**14 + q**15 -+ q**16 + q**17 Neighbours of a point in 1: 0: (1] 1 1: [16J -1 + q + q**2 + q**3 + q**4 + 2*q**5 + 2*q**6 + 2*q**7 + 2*q**8 + 2*q**9 + q**10 + q**11 + q**12 2: [10] q**9 + 0**10 + ~**11 -+ q**12 + 2*q**13 + q**14 + q**15 + 0**16 + 0**17 rle1ghoours of a 1: [10J 2: [16J 3: [lJ point in 2: 1 + q + q**2 + 4**3 + 2*q**4 + q * * 5 + q * * 6· + q* * 7 + q * * 8 -1 - 0**4 + q**~ + q**6 + q**7 + q**8 + 3*q**9 -+ 2*q**10 -+ 2*0**11 2*q**13 + q**14 -+ q**15 + q**16 q**17 + 2*q**12 + Neighbours of a point in 3: 2: [27J 1 + q + q**2 + 4**3 + 2*q**4+ 2*q**5 + 2*q**6 + 2*q**7 + 3*0**8 + 2*q**9 + 2*q**10 + 2*q**~1 + 2*q**12 + q**13 + q**14 + q**15 -+ q**16 3: [OJ -1 - q**4 - q**Q + q**9 + q**13 + q**17

***** e7t6 ***** 126 cosets 5 double cosets Sizes: 0: () [1 J 1: ( 6 ) [32J 2: (65473456) [60J 1 q + q**2 + q**3 + 2*q**4 + 2*q**5 + 3*q**6 + 3*q**7 + 3*q**8 + 3*q**9 + 3*q**10 + 3*q**11 + 2*q**~2 + 2*q**13 + 0**14 + 0**15 + q**16 q**8 + q**9 + 2*0**10 + 2*q**11+ 4*q**12 + 4*q**13 + 5*0**.4 + 5*q**15 + 6*q**16 + 6*q**17 + 5*q**18 + 5*q**~9 + 4*q**20 + 4*q**21 + 2*q**22 + 2*q**23 + q**24 + q**25 [32J q**17 + q**18 + q**19 + 2*q**20 + 2*q**21 + 3*q**22 + 3*q**,3 + 3*q**24 + 3*q**25 + 3*q**26 + 3*q**27 + 2*q**,8 + 2*q**29 + q**30 + q**31 + q**32 4: (654734562345123474563452347i23456) [lJ q**33 Neighbours of a point in 0: 1: [32J q + q**2 + q**3 + 2*q**4 + 2*q**5 + 3*q**6 + 3*0**7 + 3*q**8 + 3*q**9 + 3*q**10 + 3*q**11 + 2*q**~2 + 2*q**13 + q**14 + q**15 + q**16 Neighbours 0: [ 1 J 1: [15J of a point in 1: 1 2: [15] 3: [lJ -1 + 0 + q**2 + q**3 + 2*q**4 + . 2*q**5 + 3*q**6 + 2*0**7 + 2*q**8 + q**9 + q**10 q**7 + q**8 + 2*Q**9 + 2*q**10 + 3*q**11 + 2*q**12 + 2*q**~3 + ~**14 + q**15 q**15 Neighbours of a point in 2:' 1: [8] 1 + q + q**2 + ,*q**3 + q**4 + q**5 + q**6 2: [16] -1 - q**3 + q**. + q**5 + 2*q**6 + 3*q**7 + 3*q**8 + 3*q**9 + 2*q**10 + 2*q**11 + q**12 3: [8] q**10 + q**11 + q**12 + 2*q**13 + q**14 + q**15 + 0**16 Neighbours of a point in 3: 1: [1] 1 2: [15]

3:

[15J 4: [ 1 J Neighbours of a 3: [32] 4: [OJ q + q**2 + 2*q**3 + 2*q**4 + 3*q**5 + 2*q**6 + 2*0**7 + 0**8 + q**9 -1 - q**3 - q**~ + 0**6 + q**7 + 2*q**8. + 2*q**9 + 3*q**10 + 3*q**11 + 2*q**12 + 2*q**13 + q**14 + 0**15 q**16 point in 4: 1 + q + q**2 + ,*Q**3 + 2*0**4 + 3*q**5· + 3*q**6 + 3*Q**7 + 3*q**8 + 3*q**9 + 3*q**10 + 2*a**~1 + 2*0**12 + 0**13 + q**14 + q**15 -1 - q~*3 - ~**~ + q**11 + q**13 + q**16

***** e7,7 ***** 576 cosets 10 double cosets Sizes: 0: () [1J 1: (7) [35 J 1 q ~ q**2 + 2*q**3 + 3*q**4 + 4*q**5 + 4*q**6 + 5*q**7 + 4*q**8 + 4*0**9 + 3*q**10 + 2*q**11 + q**12 + q**13 2: {745347> [105J q**6 + 2*q**7 + 4*q**8 + 6*q**9 + 9*q**10 + 11*q**11 + 13*q**12 + 13*q**13 + 13*q**14 + 11*0**15 + 9*q**16 + 6*q**~7 + 4*q**18 + 2*q**19 + q**20 3: (74563452347> [140] q**11 + 2*q**12 + 4*q**13 + 7*q**14 + 10*q**15 + 13*q**16 + 16*q**17 + 17*0**18 • 17*q**19 + 16*q**20 + 13*q**21 + 10*q**22 + 7*q**23 + 4*q**24 + 2*0**25 + q**26

4:

<745347234512347> [7] q**15 + q**16 + q**17 + q**18 + q**19 + q**20 + q**21 5: (7453476234512347) [140J q**16 + 2*q**17 + 4*q**18 + 7*0**19 + 10*q**20 + 13*q**21 + 16*q**22 + 17*q**23 + 17*q**24 + 16*q**25 + 13*q**26 + 10*q**27 + 7*0**28 + 4*q**29 + 2*0**30 + q**31 6: (745634523474563452347) [7] 0**21 + q**22 + q**23 + 0**24 + q**25 + q**26 + q**27 7: (7456345234745634512347) [105J 0**22 + 2*q**23 + 4*q**24 + 6*q**25 + 9*q**26 + 11*q**27 + 13*q**28 + 13*q**29 + 13*q**30 + 11*Q**31 + 9*q**32 + 6*a**~3 + 4*q**34 + 2*q**35 + q**36 8: (745347623451234734562345123~7) [35J q**29 + q**30 + 2*Q**31 + 3*q**32 + 4*q**33 + 4*q**34 + 5*q**~5 + 4*q**36 + 4*q**37 + 3*q**38 + 2*q**39 + q**40 + q**41 9: (745347623451234734562345123~73456234512347) [1J q**42 Neighbours of a point in 0: 1: [35J q + q**2 + 2*q**3 + 3*q**4 + 4*q**5 + 4*q**6 + 5*q**7 + 4*q**8 + 4*q**9 + 3*0**10 + 2*q**11 + q**12 + q**13 Neighoours of a p01nt in 1: 0: [1J 1 1 : [12J 2 : [18J 3: [4J Neighbours of a 1: [6 J 2: [12J 3: [12J it: [1J 5: [4] -1 + q + q**2 + 2*0**3 + 3*q**4 + 3*q**5 + 2*q**6 + q**7 q**5 + 2*q**6 + 4*q**7 + 4*q**8 + _4*0**9 2*q**.10 + q**11 q**10 + q**11 + 0**12 + 0**13 point in 2: 1 + q + 2*q**2 • q**3 + q**it -1 - q**2 • q**~ + 2*q**4 + 4*q**5 + 3*q**6 + 3*q**7 + q**6 + q**6 + 2*q**7 + 3*0**8 + 3*q**9 + 2*q**10 + q**11 q**9 i q**10' + q**11 + q**12 + q**13

~J e i 9 h b 0 u r s 1: [1 J 2: [9 J 3: [12J of a Doint in 3: 1 5: [9 J b: [1 J 7: [3 J q + 2*q**2 + 3*~**3 + 2*q**4 + 0**5 -1 - q**2 - q**~ + q**4 + 3*q**5 + 4*q**6 + 4*q**7 + 2*0**8 + q**9 q**7 + 2*q**8 + 3*q**~ + 2*q**10 + q**11 q**10 q**11 + q**12 + q**13 Neighbours of a point in 4: 2: [15J 1 + Q + 2*q**2 + 2*0**3 + 3*q**4 + [ 0 J 5: L2QJ Neighbours 2 : [3J 3: [9J 4 : [1J 5: [12J 7 : [9J 8: [ l J Neighbours 3: [20J 6 : LOJ 7 : [15J of a of a 2*q**5 + 2*q**6 + q**7 + q**8 -1 - q**2 ~ q**~ + q**5 + q**7 + q**9 q**4 + q**5 + 2*0**6 + 3*q**7 + 3*q**8 + 3*q**9 + 3*q**1u + 2*q**11 + q**12 + q**13 point in 5: 1+ q + q**2 q**2 + 2*q**3 + 3*q**4 + 2*q**5 + q**6 q**3 -1 - 0**2 - q**~ + 2*q**5 + 3*q**6 + 5*q**7 + 3*q**8 + 2*0**9 q**8 + 2*q**9 + 3*q**10 + 2*q**11 + q**12 q**13 point in b: 1 + q + 2*q**2 + 3*0**3 + 3*q**4 + 3*q**5 + 3*q**6 + 2*q**7 + q**8 + q**9 -1 - q**2 - q**~ + q**7 + q**8 + q**10 q**5 + q**6 + 2~q**7 + 2*q**8 + 3*q**9 +. 2*q**10 + 2*q**~1 + q**12 + q**13 Neighbours 3: [4J 5: [12J 6: [ 1 ] 7: [12J of a point in 7: 8: [6 J Neighbours ot a 5: [4 J 7: [18J 8: [12J 9: [ 1 ] f\Jeighbours of a 8: [35J 9: [ 0 J 1 + q + 0**2 + y**3 q**2 + 2*q**3 + 3*q**4 + 3*q**5 + 2*q**6 + 0**7 q**4 -1 - q**2 - q**~ - q**4 + q**5 + 2*q**6 + 4*q**7 + 4*q**8 + 3*q**9 + 2*q**10 q**9 + q**10 + ~*q**11 + q**12 + q**13 point in 8: 1 + q + q**2 + 4**3 q**2 + 2*q**3 + 4*q**4 + 4*q**5 + 4*0**6 + 2*q**7 + q**8 -1 - q**2 - q**w - q**4 + 3*q**J + 3*q**8 + 4*q**9 + 3*q**10 + 2*0**11 + q**12 q**13 point in 9: 1 + q + 2*q**2 + 3*q**3 + 4*q**4 + 4*q**5 + 5*q**6 + 4*q**7 + 4*q**8 + 3*q**9 + 2*q**10 + q**11 + q**12 -1 - q**2 - q**~ - q**4 - q**6 + q**7 + q**9 + q**10 + q**11 + q**13

***** eEl,1 ***** 2lfO cosets 5 double cosets Sizes: 0: () [1J (1) [56J 1 q + q**2 + q**3 + q**4 + q**5 + 2*q**6 + 2*q**7 + 2*q**8 + 2*0**9 + 3*q**10 + 3*q**11 + 3*q**~2 + 3*q**13 + 3*q**14 + 3*q**15 + 3*q**16 + 3*Q**~7 + 3*q**18 + 3*0**19 + 2*q**20 + 2*q**21 + 2*q**~2 + 2*0**23 + q**24 + q**25 + q**26 + q**27 + 0**28 2: (123458654321) [126J q**12 + q**13 + q**14 + q**15 + 2*0**16 + 2*q**17 + 3*0**~8 + 3*q**19 + 4*q**20 + 4*q**21 + 5*q**22 + 5*0**~3 + 6*0**24 + 6*q**25 + 6*0**26 + 6*q**27 + 7*0**~B + 7*q**29 + 6*0**30 + 6*q**31 + 6*q**32 + 6*q**~3 + 5*q**34 + 5*q**35 + 4*0**36 + 4*q**37 + 3*q**~8 + 3*0**39 + 2*q**40 + 2*0**41 + q**42 + q**43 + q**44 + q**45 3: (123458675645834562345876543,1) [56J q**29 + q**30

+

0**31 + q**32 + q**33 + 2*q**34 + 2*q**~5 + 2*q*.*36 + 2*q**37 + 3*q**38 + 3*q**39 + 3*q**~0 + 3*q**41 + 3*q**42 + 3*q**43 + 3*q**44 + 3*q**~5 + 3*0**46 + 3*q**47 + 2*q**48 + 2*0**49 + 2*q**~0 + 2*q**51 + q**52 + q**53 + q**54 + q**55 + 0**56 4: (123458675645834567234561234~85674563458234561234587654321) [1J q**57 ~e1ghbours of a point in 0: 1: [56J q + q**2 + q**3 + q**4 + 0**5 + 2*q**6 + 3*q**11 3*q**16 2*q**21 q**26 + 2*q**7 + 2*q**8 + 2*q**9 + 3*q**10 + + 3*q**~2 + 3*q**13 + 3*q**14 + 3*0**15 + + 3*q**~7 + 3*q**18 + 3*q**19 + 2*q**20 + + 2*q**,2 + 2*q**23 + q**24 + q**25 + q**27 + q**28 Neighbours of a point in 1: 0: [ l J 1 1: [27J 2: [27J 3: [1J -1 + q + q**2 + c**3 + q**4 + q**5 + 2*q**6 + 2*q**7 + 2*q**8 + 2*q**9 + 3*q**10 + 2*q**.1 + 2*q**12 + 2*q**13 + 2*q**14 + q**15 + q**16 + q**17 + q**18 q**11 + q**12 + q**13 + q**14 + 2*q**15 + 2*q**16 + 2*q**.7 + 2*q**18 + 3*q**19 + 2*0**20 + 2*q**21 + 2*q**~2 + 2*q**23 + 0**24 + 0**25 + q**26 + q**27 q**28 Neighbours of a point in 2: 1: [12J 1 + q + q**2 + ~**3 + q**4 + 2*q**5 + q**6 + q**7 + 0**8 + q**9 + q**10 2: l32J -1 - 0**5 + q**o + q**7 + q**8 + q**9 + 2*q**10 T 3*q**11 + 3*q**12 + 3*q**13 + 3*q**14 + 3*q**~5 + 3*0**16 + 3*q**17 + 2*q**18 + 2*q**119 + q**20 + q**21 + q**22 3: [12J q**1~ + q**19 + 0**20 + q**21 + q**22 + 2*q*~23 + q**24 + q**25 + q**26 + q**27 + 0**28

Neighbours of a point in 3: 1: [lJ 1 2: [27J 3: [27J 4 : [ 1 J q + 0**2 + q**3 + 0**4 + 2*q**5 + 2*q**6 + 2*q**7 + 2*q**8 + 3*q**9 + 2*q**10 + 2*q**11 + 2*0**.2 + Z*q**13 + 0**14 + q**15 + q**16 + q**17 -1 - q**5 - q**~ + q**lC + q**11 + q**12 + q**13 + 2*q**14 + 2*q**15 + 2*q**16 + 2*q**17 + 3*q**.8 + 3*q**19 + 2*q**20 + 2*q**21 + 2*q**22 + 2*q**~3 + q**24 + q**25 + q**26 + q**27 q**28 Neighbours of a point in 4: 3: [56J 1 + q + q**2 + y**3 + q**4 + 2*q**5 + 2*q**6 + 2*0**7 + 2*q**8 + 3*q**9 + 3*q**10 + 3*q**.1 + 3*q**12 + 3*q**13 + 3*q**14 + 3*q**15 + 3*q**.6 + 3*0**17 + 3*q**18 + 2*q**19 + 2*q**20 + 2*q**~1 + 2*Q**22 + q**23 + Q**24 + q**25 + q**26 + q**27 4: [OJ -1 = q**5 - q**7 + q**19 + q**23 + q**28

***** e8,7 ***** 2160 cosets 10 double cosets Sizes:

c: ()

[1 ] 1: (7) [64J 1 q + 0**2 + q**3 + 2*q**4 + 2*0**5 + 2: (76584567) 3*q**6 + 4*q**7 + 4*q**8 + 4*q**9 + 5*q**10 + 5*q**11 • 5*c**~2 + 5*q**13 • 4*q**14 • 4*0**15 + 4*q**16 + 3*q**~7 • 2*q**18 + 2*q**19 + q**20 + 0**21 + 0**22 [28CJ 0**8 + q**9 + 2.0**10 + 3*0**11 + 5*0**12 + 6*0**13 + 9*q**~4 + 10*q**15 + 13*q**16 + 15*q**17 + 17*0**18 + 18*q-*15 • 20*q**20 + 20*q**21 + 20*q**22 + 20*q**23 + 18*q**24 + 17*0**25 + 15*q**26 + 13*q**27 + 10*q**28 + 9*q*.29 + 6*q**30 + 5*q**31 + 3*0**32 + 2*0**33 + q**34 + q**35 3: (76584563458234567) [448J q**17 + 2*q**18 + 3*q**19 + 5*q**20 + 7*0**21 + 10*q**22 + 14*q**23 + 17*q**24 + 20*q**25 + 24*0**26+ 27*q**27 + 30*~**28 • 32*q**29 + 32*q**30 + 32*0**31 + 32*q**32 • 30*q%*33 + 27*0**34 + 24*q**35 + 20*q**36 + 17*0**37 • 14*q**38 + 10*q**39 + 7*q**40 + 5*0**41 • 3*q**42 + 2*0**~3 • 0**44 4: (765845673456234531234567) [560J q**24 • q**25 • 2*0**26 • 4*q**27 • 6*q**28 + 8*q**29 + 12*0*-30 + 15*q**31 • 19*q**32 .+ 24*q**33 + 27*0**34 + 31*q**35 • 35*q**36 + 37*q**37 + 38*q**38 + 40*q**39 + 38*q**40 + 37*0**41 + 35*q**42 + 31*q**43 + 27*q**44 + 24*q**45 + 19*q**46 + 15*q**47 + 12*q**48 + 8*q**49 + 6*q**~0 + 4*q**51 + 2*q**52 + 0**53 + q**54 5: (765845673456234585674563458~34567) [14J q**33 + 0**34 + q**35 + q**36 + q**37 + q**38 + 2*q**39 + q**40 + q**41 + q**42 + 0**43 + 0**44 + 0**45 6: (765845673456234585674563458~234567) [448J q**34 + 2*q**35 + 3*q~*36 + 5*q**37 + 7*0**38 + 10*0**39 + 14*q**40 + 17*0**41 + 20*q**42 + 24*q**43 + 27*q**44 + 30*q**45 + 32*q**46 + 32*q**47 + 32*0**48 + 32*q**49 + 30*q-*50 + 27*q**51 + 24*q**52 + 20*0**53 + 17*0**54 • 14*q**55 + 10*q**56 + 7*q**57 + 5*0**58 + 3*0**59 + 2*q**~0 + q**61 7: (765845634587234561234584567~456234581234567) [280J q**43 + q**44 + 2*q**45 + 3*0**46 + 5*q**47 + 6*q**48 + 9*q**~9 • 10*q**50 + 13*q**51 + 15*q**52 + 17*q**53 + 18*q**54 + 20*0**55 + 20*q**56 + 20*0**57 + 20*q**58 + 18*q**59 + 11*q**60 + 15*0**61 + 13*q**62 + 10*q*~63 • 9*q**64 + 6*q**65 + 5*q**66 • 3*q**67 + 2*q**68 + q**69 + q**70 8: (765845634587234561234584567~4562345845673456234581234567) [64J q**56 + q**57 + q**58 + 2*q**59 + 2*q**60 + 3*q**61 + 4*q**02 + 4*0**63 + 4*q**64 + 5*0**65 + ~*q**66 + 5*q**07 + 5*0**68 + 4*q**69 + 4*0**70 + 4*q**71 + 3*0**,2 + 2*q**73 + 2*q**74 • q**75 + q**76 + 0**77 '3: ( 7 65

a

4 5 6 7 3 '+ 5 6 2 3 4 5 8!5 6 7 '+ 56 1 234 J 8£' 723'+ 5 612 345 83 '+ 56 7 2 3 456 1 23'+ 5 8 45 673456234581234567) [IJ q**78

Neighbours of a point in 0: 1: [64J q + q**2 + q**3 + 2*0**4 + 2*q**5 + 3*q**6 + 4*0**7 + 4*0**& + 4*0**9 + 5*q**10 + 5*q**11 + 5*0**.2 + 5*q**13 + 4*q**14 + 4*q**15 + 4*q**16 + 3*q**~7 + 2*q**18 + 2*q**19 + q**20 + q**21 + q**22 ~e1Qhbours of a point in 1: 0: [ l J 1 1: [21J -1 + q + q**2 + q**3 + 2*q**4 + 2: [35J 3: [7 ] 2*q**5 + 3*q**6 + 3*q**7 + 3*q**8 + 2*q**9 + 2*q**10 + q**11 + 0**12 0**7 + q**8 + 2*0**9 + 3*q**10 + 4*q**11 + 4*0**12 + 5*q**~3 + 4*q**14 + 4*q**15 + 3*q**16 + 2*q**17 + q**18 + q**19 q**16 + q**17 + q**18 + 0**19 + q**20 + q**21 + 0**22 Neighbours of a point in 2: 1: [8J 1 + a + q**2 + ~*0**3 + q**4 + 0**5 + q**6 2 : [24 J 3 : [24J 4 : [8J Neighbours 1 : [1J ,.,

.

_{[15 J} c: • 3: [21J 4: [20J c:: • '-'

.

[ l J [6J of a rJeighbours of a 2: [4 J 3: [16 J I.. r ... "., "'t. L.~"'t.J 6: [16J [4 J -1

-

q** 3 + q**'t + q**5 + 2*q**6 + 4*q**7 + 4*0**8 + 4*'1**9 + 4*q**10 +'3*0**11 2*q**12 + q**13 0**10 + 2*q**11 + 3*q**12 + 4*c;**13 + 4*0.**14 4*q**15 + 3*0**~6 + 2*0**17 + q**18 q**16 + q**17 + q**18 + 2*q**19 + q**20 + q**21 + point ~-_{I I I} 1 q + q**2 2*q**6 + q**22 '2 • '-'

.

+ 2*q**3 2*q**7 + + 2*q**4 + 3*0**5 + q**8 + 0**9 -1 - q**3 - q**~ + q**6 + 2*q**7 + + + 3*q**8 + 3*q**9 + 4*q**10 + 4*q**11 + 3*q**12 + 2*q**13 + q**14 + q**15 q**10 + q**ll + 2*q**12 + 3*q**13 +'3*q**14 + 3*q**15 + 3*q**~6 + 2*q**17 + q**18 + q**19 q**16 q**17 + 0**18 + q**19 + q**20 + q**21 + 0**22 point in 4: 1 + q + q**2 + y**3 q**3 + 2*q**4 + 3*q**5 2*q**8 + q**9 ~1 = q**3 ~ q**~ - 0**6 + q**7 + 2*0**8 + 3*q**9 + 5*0**10 + 5*q**11 + 5*q**12 + 4*q**13 + 2*0**.4 + 0**15 0**13 + 2*q**14 + 3*q**15 + 4*0**16 + 3*q**17 + 2*q**18 + q**19 0**19 + q**20 + q**21 + q**22 ~e1ghbours of a point in 5: 3: [32J 1 + 0 + q**2 + ,*0**3 + ~*q**4 + 5 : 6 : [ 0 J [32 J 3*q**5 + 3*q**6 + 3*q**7 + 3*q**8 + 3*q**9 + 3*q**10 + 2*q**~1 + 2*q**12 + q**13 + q**14 + q**15 -1 -q**3 - q**~ + q**11 + q**13 + q**16 0**7 + 0**8 + q**5 + 2*q**10 + 2*q**11 + 3*q*~12 + 3*0**~3 + 3*q**14 + 3*q**15 + 3*q**16 + 3*0**11 + 2*q**~B + 2*q**19 + q**20 + q**21 + q**22

~,eighbours 3 : [ 6 ] 4 : [20J c; • _{[ l J} "-'

.

6 : [21J 7: [1 ~ J 8 : [lJ Neighbours 4 : [8J 6 : [24J 7 : [24J 8 : [8J of a 01 a point in

6:

1 + c + q**2 + y**3 + q**4 + 0**5 q**3 + q**4 + 2-0**5 + 3*0**6 + 3*q**7 + 3*q**8 + 3*q**9 + 2*0**10 + q**11 + q**12 q**q -1 - q**3 - q**~ - q**6 + q**7 + q**8 + q**9 + 3*0**10 + 4*q**11 + 4*q**12 + ~*q**13 + 3*q**~4 + ~*0**15 + 2*q**16 q**13 + 0**14 + 2*0**15 + 2*q**16 + 3*q**17 + 2*q**18 + 2*q**.9 + 0**2C + 0**21 q**22 point in 7: 1 + q + q**2 + ~*0**3 + q**4 + q**5 + 0**6 q**4 + 2*q**5 + 3*0**6 + 4*q**7 + 4*q**8 + 4*q**9 + 3*q**lu + 2*0**11 + q**12 -1 - q**3 - q**~ - q**6 + 2*q**10 + 3*q**11 + 4*q**.2 + 5*q**13 + 4*q**14 + 4*q**15 + 3*q**16 + 2*q**~7 + q**18 q**16 + q**17 + 0**18 + 2*q**19 + q**20 + q**21 + q**22 Neighbours of a point in 8: 6: [7J 1 + q + q**2 + y**3 + q**4 + q**5 + q**6 7: [35J q**3 + q**4 + 2*0**5 + 3*q**6 + 4*q**7 + 8: [21J [1J 4*q**8 + 5*0**9 + 4*q**10

+

4*0**11 + 3*q**12 + 2*q**13 + q**14 + q**15 -1 - 0**3 - q**~ - q**6 - q**9 + q**10 + 0**11 + 2*q**12 + 3*q**13 + 3*0**14 + 3*q**15 + 4*q**.6 + 3*0**17 + 2*q**18 + 2*0**19 + q**20 + q**21 q**22 ~eighbours of a point in 9: 8: [64J 1 + q + q**2 + ~*q**3 + 2*q**4 + 3*q**5 + 4*0**6 + 4*q**7 + 4*q**8 + 5*q**9 + 5*0**10 + 5*q**~1 + ~*q**12 + 4*q**13 + 4*q**14 + 4*q**15 + 3*q**~6 + 2*q**17 + 2*q**18 + q**19 + q**20 + q**21 9: [OJ -1 - q**3 - q**~ - 0**6 - q**9 + q**13 + 0**16 + q**17 + q**19 + q**22

***** e8,8 ***** 17280 cosets 35 doubLe cosets Sizes: 0: () [1J 1: (8) [56J 1 q + q**2 + 2*q**3 + 3*q**4 + 4*q**5 + 5*q**6 + 6*q**7 + 6*q**8 + 6*q**9 + 6*q**10 + 5*q**11 + 4*q**~2 + 3*q**13 + 2*q**14 + 0**15 + q**16 2: (856458) [280J q**6 + 2*q**7 + 4*q**8 + 7*0**9 + 11*q**10 +

...

'-'

.

4: 5 : 6 : 7: 8 : 10: 11 : 15*q**11 + 20*q**12 + 24*q**13 + 27*~**14 + 29*0**15 + 29*q**16 + 27*q**17 + 24*q**18 + 20*0**19 + 15*0**20 + 11*q**21 + 7*q*-22 + 4*0**23 + 2*q**24 + q**25 (85674563458) [560J q**ll + 2*q**12 + 5*q**13 + 9*q**14 + 15*q**15 + 22*0**16 + 31*q**17 + 39*q**18 + 47*q**19 + 53*q**20 + 56*q**21 + 56*q**22 + 53*q**23 + 47*q**24 ~ 39*0**25 + 31*q**26 + 22*q**27 + 15*q**28 + 9*q**29 + 5*0**30 + 2*q**31 + q**32 (856458345623458) [56J q**15 + 2*q**16 + 3*0**17 + 4*q**18 + 5*q**19 + 6*q**20 + 7*q**~1 + 7*q**22 + 6*0**23 + 5*q**24 + 4*q**25 + 3*q**,6 + 2*q**27 + 0**28 (8564587345623458) [1120J q**16 + 3*q**17 + 7*0**18 + 14*q**19 + 24*q**20 + 37*q**21 + 53*q**22 + 70*q**23 + 86*q**24 '+ 100*q**25 + 109*q**26 + 112*q**27 + 109*q**28 + 100*0**29 + 86*q**30 + 70*q**31 + 53*q**32 + 37*q**33 + 24*q**34 + 14*0**35 + 7*q**36 + 3*q**~7 + q**38 (856745634585674563458) [28J q**21 + q**22 + 2*q**23 + 2*q**24 + 3*q**25 + 3*q**26 + 4*q**~7 + 3*q**28 + 3*q**29 + 2*0**30 + 2*q**31 + q**32 + q**33 (8564583456723456123458) [28CJ q**22 + 2*q**23 + 4*q**24 + 7*q**25 + 11*q**26 + 15*q**27 + 20*q**28 + 24*q**29 + 27*q**30 + 29*q**31 + 29*q**32 + 27*q**33 + 24*q**34 + 20*q**35 + 15*q**36 + 11*q**37 + 7*q**38 + 4*q**39 + 2*q**40 + 0**41 (8567456345823456123458) [280J q**22 + 2*q**23 + 4*0**24 + 7*q**25 + 11*q**26 + 15*0**27 + 20*q**28 + 24*q**29 + 27*0**30 + 29*q**31 + 29*q**32 + 27*Q**33 + 24*q**34 + 20*0**35 + 15*0**36 + 11*q**37 + 7*q**38 + 4*q**39 + 2*q**40 + q**41 (8567456345856745623458) [ 8 4 0 J q * * 2 2 + 3*0**23 + 7*0**24 + 13*q**25 + 22*q**26 + 33*q**27 + 46*q**28 + 59*q**29 + 71*q**30 + 80*q**31 + 85*q~*32 + 85*q_*33 + 80*0**34 + 71*q**35 + 59*q**36 + 46*q**37 + 33*q**38 .. 22*q**39 + 13*0**40 + 7*q**41 + 3*q**42 + q**43 (8567456345856723456123458) [1680J q**2S + 3*q**26 + 8*q**27 + 16*q**28 + 29*q**29 + 46*q*:*30 + 68*q* *31 .. 92*q**32 + 117*q**33 + 139*q**34 + 156*q~*35 + 165*q**36 + 165*q**37 + 156*0**38+ 139*0**39 + 117*q**40 + 92*y**41 + 68*0**42 + 46*q**43 + 29*q**44 + 16*q~*45 + 8*q**46 + 3*q**47 + q**48 (85645873456234584567345623~58) [280J q~*29 + 2*q**30 + 4*0**31 + 7*0**32 + 11*0**33 + 15*q**34 + 20*q-*35 + 24*q**36 + 27*q**37 + 29*0**38 .. 29*q**39 + 27*q**40 + 24*0**41 + 20*q**42 + 15*q**43 + 11*0**44 + 7*q*=45 + 4*q**46 + 2*Q**47 ~ q**48

12: (8564587345b234584567345612~458) [1680J q-*30 + 3*q**31 + 8*q**32 + 16*q**33 + 29*q**34 + 46*q**35 + 68*qx*36 + 92*q**37 + 117*0**38 + 139*q**39 + 156*q**40 + 165-q**41 + 165*q**42 + 156*0**43 + 139*q**44 + 117*q**45 + 92*~**46 + 68*q**47 + 46*q**48 + 29*q**49 + 16*q**50 + 8*q*-51 + 3*q**52 + q**53 13: (85674563458567456345823456~23458) [168J 0**32 + 2*q**33 + 4*q**34 + 6*q**35 + 9*q**36 + 12*q**37 + 15*qa*38 + 17*0**39 + 18*q**40 + 18*q**41 + 17*q**42 + 15*q**43 + 12*0**44 + 9*q**45 + 6*0**46 + 4*q**47 + 2*0**~8 + 0**49 14: (85645834567234561234585674~63458) [168J q**32 + 2*0**33 + 4*q**34 + 6*q**35 + 9*q**36 + 12*q**37 + 15*q-*38 + 17*q**39 + 18*q**40 + 18*0**41 + 17*0**42 + 15*q-*43 + 12*q**44 + 9*q**45 + 6*q**46 + 4*q**47 + 2*q**~8 + q**49 15: (85674563458567456234586723~56123458) [1120J q**35 + 3*q**36 + 7*q**37 + 14*0**38 + 24*q**39 + 37*q**40 + 53*q**41 + 70*0**42 + 86*q**43 + 100*q**44 + 109*q**45 + 112*0**46 + 109*q**47 + 100*q**48 + 86*q**49 + 70*q**50 + 53*q**51 + 37*q**52 + 24*q**53 + 14*q**54 + 7*q**55 + 3*q**~6 + 0**57 16: (856458345672345b1234584567~45623458) [1120J q**35 + 3*q**36 + 7*q**37 + 14*q**38 + 24*q**39 + 37*q**40 + 53*q-*~1 + 70*q**42 + 86*q**43 + 100*q**44 + 109*q**45 + 112*q**46 + 109*0**47 + 100*q**48 + 86*q**49 + 70*q**50 + 53*q-*51 + 37*0**52 + 24*q**53 + 14*0**54 + 7*q**5~ + 3*q**06 + q**57 17: (85674563458234561234583456/23456123458) [70J q**38 + q**39 + 2*q**40 + 3*q**41 + 5*q**42 + 5*q**43 + 7*q**~4 + 7*q**45 + 8*q**46 + 7*q**47 + 7*q**48 + 5*q**~9 + 5*q**50 + 3*q**51 + 2*q**52 + q**53 + q**54 18: (85674563458567234561234585D723456123458) [1680J q**39 + 3*q**40 + 8*q**41 + 16*q**42 + 29*q**43 + 46*0**44 + 68*q-*45 + 92*q**46 + 117*q**47 + 139*q**48 + 156*q**49 + 165 x q**50 + 165.0**51 + 156*q**52 + 139*0**53 + 117*0**54 + 92*4**55 + 68*q**56 + 46*q**57 + 29*q**58 + 16*q**59 + 8*q**60 + 3*q**61 + 0**62 19: (85645873456234584567~45623~584567345623458) [8J 0**42 + q**43 + q**44 + q**45 + q**46 + q**47 + q**48 + q**49 20: (856458345672345b1234585674~6345823456123458) [8J q**43 + 0**44 + q**45 + q**46 + q**47 + q**48 + q**49 + q**50 21: (85645873456234584567345623~5845673456123458) [168J 0**43 + 2*q**44 + 4*0**45 + 6*q**46 + 9*0**47 + 12*q**48 + 15*q**49 + 17*q**50 + 18*q**51 + 18*q**52 + 17*q**53 + 15*q**54 + 12*q**55 + 9*q**56 + 6*q**57 + 4*q**58 + 2*q**~9 + q**60 22: (85645873456234584567345612~4584567345623458) [168J q**43 + 2*q**44 + 4*0**45 + 6*q**46 + 9*q**47 + 12*q**48 + 15*q**49 + 17*Q**50 + 18*0**51 + 18*q**52 + 17*q**53 + 15*q-*54 + 12*q**55 + 9*0**56 + 6*0**57 + 4*q*~58 + 2*q**~9 + q**60 23: (856458734562345g4567345612~45845673456123458) [1680J q**44 + 3*q**45 + 8*q**46 + 16*q**47 + 29*0**48 + 46*q~*49 + 68*q**50 + 92*0**51 + 117*q**52 + 139*~**53 + 156*q**54 + 165*q**55 + 165*q**56 + 156*0**57 + 139*q**58 + 117*~**59 + 92*y**60 + 68*q**61 + 46*q**62 + 29*q**63 + 16*q**64 + 8*q**65 + 3*q**66 + q**67

24: (85645834567234561234585674~63458723456123458) [280J q**44 + 2*q**45 + 4*q**46 + 7*q**47 + 11*0**48 + 15*q**49 + 20*q**50 + 24*q**51 + 27*q**52 + 29*q**53 + 29*q**54 + 27*q**55 + 24*q**56 + 20*0**57 + 15*q**58 + 11*0**59 + 7*q**60 + 4*0**61 + 2*q**62 + q**63 25: (85645834567234561234584567~4562345856723456123458) [840J .q**49 + 3*0**50 + 7*q**51 + 13*q**52 + 22*q**53 + 33*q**54 + 46*q**55 + 59*0**56 + 71*0**57 + 80*0**58 + 85*a**59 + 85*q**60 + 80*q**61 + 71*q**62 + 59*q**63 + 46*0**64 + 33*q**65 + 22*0**66 + 13*0**67 + 7*0**68 + 3*q**69 + 0**70 26: (85674563458567456234586723~561234583456723456123458) [280J q**51 + 2*q**52 + 4*q**53 + 7*q**54 + 11*q**55 + 15*q**56 + 20*q**57 + 24*0**58 + 27*q**59 + 29*0**60 + 29*0**61 + 27*q**62 + 24*q**63 + 20*a**64 + 15*0**65 + 11*0**66 + 7*0**67 + 4*0**68 + 2*0**69 + q**70 27: (85674563458234561234583456/234561234584567345623458) [280J q**51 + 2*q**52 + 4*Q**53 + 7*q**54 + 11*q**55 + 15*q**56 + 20*q**57 + 24*q**58 + 27*q**59 + 29*0**60 + 29*0**61 + 27*q**62 + 24*q**63 + 20*0**64 + 15*q**65 + 11*0**66 + 7*0**67 + 4*q**68 + 2*q**69 + q**70 28: (856745634585672345612345850723456123458456723456123458) [1120J q**54 + 3*q**55 + 7*q**56 + 14*0**57 + 24*q**58 + 37*q**59 + 53*0**60 + 70*q**61 + 86*0**62 + 100*q**63 + 109*q**64 + 112*q**65 + 109*q**66 + 100*0**67 + 86*q**68 + 70*q**69 + 53*0**70 + 37*0**71 + 24*0**72 + 14*q**73 + 7*0**74 + 3*q**J5 + q**76 29: (85645873456234584567345612~45845673456234583456723456123458) [28J q**59 + q**60 + 2*q**61 + 2*q**62 + 3*q**63 + 3*q**64 + 4*q**u5 + 3*~**66 + 3*q**67 + 2*0**68 + 2*0**69 + q**70 + 0**71 30: (85645873456234584567345612~458456734561234583456723456123458) [560J q**60 + 2*q**61 + 5*q**62 + 9*q**63 + 15*0**64 + 22*q**65 + 31*q**66 + 39*q**67 + 47*q**68 + 53*0**69 + 56*0**70 +-56*q**71 + 53*q**72 + 47*q**73 + 39*q**74 + 31*q**75 + 22*q**76 + 15*q**77 • 9*0**78 • 5*q**79 + 2*q**80 + q**81 31: (85674563458567456234586723~56123458345672345612345845673456 23458) [56J 0**64 + 2*0**65 + 3*0**66 + 4*q**67 + 5*q**68 + 6*q**69 + 7*q**IO + 7*q**71 + 6*0**72 + 5*q**73 + 4*0**74 + 3*q**/5 + 2*q**76 + q**77 32: (856?4563458567456234586723~56123458345672345612345834567234 56123458) [280 J q**67 + 2*q**68 + 4*q**69 + 7*q**70 + 11*q**71 + 15*q**72 + 20*q**73 + 24*q**74 + 27*q**75 + 29*0**76 + 29*q**77 + 27*q**78 + 24*q**79 ~ 20*q**80 + 15*0**81 + 11*q**82 + 7*q**83 + 4*q**84 + 2*q**85 + 0**86 33: (85645873456234584567345612~45845673456123458345b72345612345 834567234561234~8) [56J 0**76 + 0**77 + 2*q**78 + 3*q**79 + 4*0**80 + 5*q**81 + 6*q**v2 + 6*q**83 + 6*0**84 + 6*q**85 + 5*0**86 + 4*q**07 + 3*q**88 + 2*0**89 + q**90.+ q**91 34: (8564587345623458A567345612~45845673456123458345672345 612345 B34567234561234~83456723456123458) [1J q *

*

'3 2,

~eighbours of a point in 0: 1: [56J q + q**2 + 2*q*-3 + 3*q**4 + 4*0**5 + 5*q**6 + 6*q**7 + 6*0**& + 6*q**9 + 6*q**10 + 5*q~*11 + 4*0**~2 + 3*0**13 + 2*0**14 + q**15 + 0**16 ~e;ghbours of a point in 1: 0: [1J 1 1: [15J -1 + q + q**2 + 2*q**3 + 3*q**4 + 3*~**5 + 3*0**6 + 2*q**7 + q**8 2: [30J q**5 + 2*q**6 + 4*q**7 + 5*q**8 + 6*q**9 + 5*q**10 + 4*q**~1 + 2*q**12 + q**13 3: [10J q**10 + q**11 + 2*q**12 + 2*0**13 + 2*q**14 + q**15 + q**16 Npighbours of a point in 2: 1: [6J 1 + q + 2*q**2 ~ q**3 + 0**4 2: [16J -1 - q**2 + q**~ + 2*q**4 + 4*q**5 + 3: UBJ 4: [3 J 5: [12J B: [ l J Neighbours 1 • r 1 -, ...

.

L ... .J 2: [9J 3: [15J 5: [18J 6: [ 1J 7 : [3J 9 : [6J 10: [3 J 4*q**6 + 4*q**7 + 2*q**8 + q**9 q**6 + 2*q**7 + 4*q**& + 4*q**9 + 4*q**10 + 2*q**11 + q**12 q**9 + q**10 + ~**11 q**10 + 2*q**11 + 3*q**12 + 3*q**13 + 2*q**14 + q**15 q**16 of a point in 3: 1 q + 2*q**2 + 3*~**3 + 2*q**4 + Q**5 -1 - q**2 - q**~ + q**4 + 3*q**5 + 5*q**6 + 5*q**7 + 3*q**8 + q**9 q**7 + 3*q**8 + 5*q**9 + 5*Q*~10 + 3*q**11 + q**12 q* *1 0 q**11 + q**12 + 0**13 0**11 + 2*q**12 + 2*Q**13 + 0**14 q**14 + q**15 + 0**16 Neighbours of a point in 4: 2: [15] 1 + q + 2*q**2 ~ 2*q**3 + 3*q**4 + 2*q**5 + 2*0**6 + q**7 + 0**8

4:

[6J 5: [20J 8: t1.5J I\;eighbours 2: [3] 3: ['3J 4: [1 J 5: [15] 7 : [3J 8 : [3J 9 : [S'J 10: [9 J 11: [ l J 12: [3J -1 - q**2 - q**~ + q**5 + q**6 + 2*q**7 + 0**8 + 2*0**9 + 0**10 + q**11 q**4 + q**5 + 2*q**6 + 3*q**7 + 3*q**8 + 3*q**9 + 3*q**lu + 2*0**11 + q**12 + 0**13 q**8 + q**9 + 2w_q**10 + 2*q**il + 3*q**12 + 2*q**13 + 2*q**~4 + q**15 + q**16 of a point in 5: 1 + q + q**2 q**2 + 2*q**3 + 3*q**4 + 2*q**5 + q**6 q**3 -1 -q**2 - o**~ + 2*q**5 + 4*q**6 + 6*q**7 + 4*q**8 + 2*q**9 q**9 + q**10 + ~**11 q**8 + 0**9 + qa~10 q**8 + 2*0**9 + 3*q**10 + 2*q**11 + q**12 q**16 + 2*q**11 + 3*0**12 + 2*q**13 + q**14 q**13 q**14 + q**15 + q*~16